Nth Fibonacci Number Calculator
The Fibonacci sequence starts with 0 and 1, and every term after that is the sum of the two terms before it, giving 0, 1, 1, 2, 3, 5, 8, 13 and on. This calculator returns the nth Fibonacci number for a position you choose, counting from F(0) equal to 0 and F(1) equal to 1. It computes the term by adding successive pairs, the exact integer recurrence that defines the sequence, so the answer is always a whole number with no rounding. The Binet closed-form expression, which involves the golden ratio, gives the same value and explains why Fibonacci numbers grow at a steady geometric rate, but the iterative sum is used here to keep large terms exact. The sequence appears throughout mathematics and nature, from the branching of plants and the spiral of shells to algorithms and the golden ratio that successive Fibonacci ratios approach. Students, programmers and puzzle solvers use it constantly. Enter the position n to get the Fibonacci number immediately; the position is a small whole number, so it is left as a plain count. Every figure here is computed deterministically from the definition shown below, with a worked example that reconciles exactly to the calculator so you can follow each step yourself.
Each Fibonacci term is the sum of the two before it, starting 0, 1. The 10th Fibonacci number (counting F(0) = 0) is 55.
Nth Fibonacci Number formula
F(n) = F(n-1) + F(n-2)
F(0) = 0, F(1) = 1
Binet: F(n) = (phi^n - psi^n) / sqrt(5)
phi = (1 + sqrt 5) / 2 (golden ratio)
psi = (1 - sqrt 5) / 2
Each term is the sum of the two preceding terms. The Binet formula gives the same value in closed form using the golden ratio, but the iterative sum keeps the result an exact integer.
Worked example
Find the 10th Fibonacci number, with F(0) defined as 0.
- Build up: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
- So F(8) = 21, F(9) = 34
- F(10) = 34 + 21 = 55
The 10th Fibonacci number is 55. This is the calculator's default input, so the result above matches the widget exactly.
First Fibonacci numbers
F(n) with F(0) = 0.
| n | F(n) |
|---|---|
| 5 | 5 |
| 8 | 21 |
| 10 | 55 |
| 12 | 144 |
| 15 | 610 |
Integer-sequence reference: US National Institute of Standards and Technology (NIST).
Nth Fibonacci Number Calculator: frequently asked questions
What is the Fibonacci sequence?
It is the sequence that begins 0 and 1, with every later term equal to the sum of the two terms before it: 0, 1, 1, 2, 3, 5, 8, 13, 21 and so on. It is one of the most famous integer sequences in mathematics and appears in many natural growth patterns.
Does counting start at 0 or 1?
This calculator uses the common convention F(0) equal to 0 and F(1) equal to 1, so the position counts from zero. With that indexing the 10th Fibonacci number is 55. Some texts start at F(1) equal to 1, which shifts the index by one; check which convention a source uses.
What is Binet's formula?
Binet's formula gives the nth Fibonacci number directly using the golden ratio, as phi to the n minus psi to the n, all divided by the square root of five. It produces the same integers as the recurrence, and it shows that Fibonacci numbers grow geometrically at the golden ratio.
How are Fibonacci numbers linked to the golden ratio?
The ratio of consecutive Fibonacci numbers gets closer and closer to the golden ratio, about 1.618, as the numbers grow. That is why the golden ratio appears in Binet's formula and why Fibonacci spirals look so similar to golden spirals.
What is the Fibonacci rule?
Each term equals the sum of the two before it, F(n) = F(n-1) + F(n-2), starting from F(0) = 0 and F(1) = 1. Building up the sequence, the 10th term is 34 plus 21, which is 55.
Official sources
- Mathematical functions and integer-sequence reference data (Digital Library of Mathematical Functions): US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.